Kumar Raman has written a helpful comment on an earlier post where I made a conjecture which rules out spin liquid ground states for wide classes of Heisenberg antiferromagnetic spin models.

He points out that one can construct SU(2) invariant models with ground states similar to the spin liquid quantum dimer models, but concedes these models are not very physical.

Section 1.10 of Roderich Moessner and Kumar's review is a helpful description

of the procedure to construct a model.

So I have some questions?

Is the claim that the Hamiltonian (1.36) may have a spin liquid ground state?

In light of these points, how does my conjecture need to be sharpened to rule out such models?

How about no terms in Hamiltonian beyond first- and second-neigbour interaction and single plaquette, and first-neighbour must have the largest interaction?

**Conjecture**: Consider a spin-1/2 Heisenberg model on a two-dimensional lattice with short range antiferromagnetic exchange (both pairwise and ring exchange are allowed) interactions where the nearest neigbour interaction is the strongest. The Hamiltonian is invariant under SU(2) x L, where L is a space group and there is a total of spin-1/2 in the unit cell. Then the ground state spontaneously breaks at least one of the two symmetries SU(2) and L.

Dear Ross,

ReplyDeleteMany apologies for the delayed response.

The claim is the following:

1. Pick any lattice.

2. Decorate the lattice by adding an even number N of sites to each link, as done in Fig 1.12 for the square lattice.

3. Write down the analog of Hamiltonian 1.39 for the decorated lattice. This is an SU(2) invariant Hamiltonian which will contain a nearest-neighbor interaction as well as multiple spin interactions.

4. Perturb that Hamiltonian with the analog of Eq. 1.38 for the lattice you are working with. Note that this is also an SU(2) invariant operator.

The claim: The ground state phase diagram of this SU(2) invariant spin model will, in the limit of large decoration N, correspond exactly (modulo certain details which are not important here) to the ground state phase diagram of the quantum dimer model of the undecorated lattice chosen in step 1. As shown in Fig. 1.8, choosing a non-bipartite lattice will give you a spin liquid phase situated between valence bond crystals.

A couple points to note: (1) The two states of a link (i.e. occupied by a dimer vs. empty) in the dimer model correspond to the two dimerizations of the decorated link in the spin model. (2) As noted in the given reference, the error term becomes exponentially small with the number of decorated sites. Therefore, it is likely that this procedure will work at relatively small values of N.

If you want to rule out such models, then one way might be to restrict the interactions in the manner you suggest. Another way would be to require the model to be defined on, say, a Bravais lattice. However, while a simpler model would stand a greater chance of being physically realized, any restriction that comes to my mind seems a bit arbitrary. I'm not sure what issue of fundamental principle regarding the existence of spin liquids remains unsettled.

As you point out in part 3 of this series (your blog entry on Feb. 5, 2010), there is strong numerical evidence for a spin liquid on the triangular Heisenberg model with ring exchange. However, because it is not possible to numerically distinguish a spin liquid from a complicated type of long range order (which, as the dimer models suggest, is often what happens), we were looking for an analytical argument. This is what led us to consider the complicated lattices discussed above.

With best regards,

Kumar